Find a quadratic polynomial whose zeros are $5 + \sqrt 2$ and $5- \sqrt 2$.
- $k \{x^2 - 2 x + 23 \}$, where $k$ is any non-zero real number.
- $k \{x^2 - 16 x + 23 \}$, where $k$ is any non-zero real number.
- $k \{x^2 - 10 x + 23 \}$, where $k$ is any non-zero real number.
- $k \{x^2 - 14 x + 23 \}$, where $k$ is any non-zero real number.
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C
Correct answer
Explanation
Sum of zeros = (5 + sqrt(2)) + (5 - sqrt(2)) = 10. Product of zeros = (5 + sqrt(2))(5 - sqrt(2)) = 25 - 2 = 23. Polynomial is k(x^2 - (sum)x + product) = k(x^2 - 10x + 23).
AI explanation
For any quadratic polynomial, the coefficient of x is the negative sum of the zeros and the constant term is the product of the zeros. The sum of the given zeros, (5 + sqrt(2)) and (5 - sqrt(2)), is 10, and their product is 5^2 - (sqrt(2))^2, which equals 25 - 2 = 23. Placing these values into the standard form x^2 - (sum)x + (product) gives x^2 - 10x + 23. Including a non-zero constant multiplier k yields the general polynomial k(x^2 - 10x + 23).